33. SOME SERIES 



^" = l+*+f, +^ + --- [33a] 



sina;=a; +— ... [33b] 



3! 5! 7! 



x"- x^ a;" 

 cos^ = l-- +---... [33c] 



tan a; = a; + A a;'^ + -?- ar^ + -1^ x"^ . . . \x\ <^ [33dl 



3 15 315 2 



cot a; =1 -^ -X x^ --^ x^ . . . [la;! < v\ [33e] 



a; 3 45 945 



X X ^ 



sinh X ^ x + — + — + — , . . fasn 



3! 5! 7! ^ ^ 



3.2 JJ.4 ^6 



cosh a; = l+ — + — + — ... [332] 



2! 4! 6! ^ 



tanh X = X -^ x^ + ^ x^ - J^ a;' . . . kl < iL [33h] 



3 15 315 2 



coth ar =l + £-i a;^ + J_ a;^ . . . [|a;| < rr ] [33i] 



a; 3 45 945 . 



sin~^ a; = a; + J_ a;'' + ^'^ x^ + I'^'S x'' . . . [\x\ < 1] [33i] 



2-3 2-4-5 2-4-6-7 



tan-i X = X -Ix^ +1 x^ -i-ar' . . . [\x\ < 1] 

 3 5 7 



= '' -i+ ^-—4-- - - [1^1 >1] 

 2 a; 3^ 5a; 5 ' ' 



[33k] 



sinh-i a; = ar -J_ x^ + -1^ a;^ . . . [|a;| < 1] [331] 



2-3 2-4-5 



tanh"! X = X +lx^ +l.x^ +l.x'^ + . . . ar'^ + . . . [\x\ < 1] [33m] 



3 5 7 



In (1 +a;) = a; -la;2 +1 x^ -Lx"^ . . . V\x\ < 1] [33n] 



2 3 4 



Except as indicated, the expansions hold for all values of x. See Smithsonian Mathematical 

 Formulas!^ for other series. 



69 



